The Yau-Tian-Donaldson conjecture for anti-canonical polarization was
recently solved affirmatively by Chen-Donaldson-Sun and Tian. However,
this conjecture is still open for general polarizations or more
generally in extremal Kähler cases. In this book, the unsolved cases of
the conjecture will be discussed.
It will be shown that the problem is closely related to the geometry of
moduli spaces of test configurations for polarized algebraic manifolds.
Another important tool in our approach is the Chow norm introduced by
Zhang. This is closely related to Ding's functional, and plays a crucial
role in our differential geometric study of stability. By discussing the
Chow norm from various points of view, we shall make a systematic study
of the existence problem of extremal Kähler metrics.